Proposition 1 [1]}
Let \(g_{\mathcal {X}}(\cdot )\)  be the Frank-Wolfe gap defined in (REF ) and \(\mathcal {G}_{\mathcal {X}}(\cdot )\)  be the gradient mapping defined in (REF ). Then, we have
\(\Vert  \mathcal {G}_{\mathcal {X}}(x, \nabla F(x), \beta )\Vert ^2 \le g_{\mathcal {X}}(x, \nabla F(x)),\qquad \forall x\in \mathcal {X}.\) 
